In Complex Analysis one defines: $$e^{i\theta}:=\cos \theta+i\sin\theta$$ and proves identities like $e^{i\theta}e^{i\phi}=e^{i(\theta+\phi)}$ etc, using basic trigonometry.
One also defines complex exponential function, call it, $E(\cdot)$, such that: $$E(x+iy):=e^{x}e^{iy}=e^x\cos y+i e^x\sin y.$$ Like above, one proves identities like $E(z)E(w)=E(z+w)$ etc. It is interesting to note that $E(x+i0)=e^x$ so $E$ extends in the complex plane the real exponential funtion (so one may also extend the notation like $E(z)=e^z$ for $z\in \mathbb{C}$).
I would like to point out the use of "e" on the above notations.
As far as I understand, and please feel free to correct me if I am wrong, the only place where $e$ is the famous Euler's constant is on $e^x$ in the definition of $E(\cdot).$ It seems to me that it is all the nice properties satisfied that make us use a base-power notation like $e^{i\theta}$ and $e^z$, instead of somehing like: $$\varpi(\theta) ~~~\mathrm{or} ~~~\exp_\mathbb{C}(z),$$ respectively (in addition to the fact that $\exp$ works great in the real case, so we would like to extend t h i s function on the complex plane, instead of extending, for example, $2^x$). If so of course, from that point of view, one could say that "the Most Beautiful Mathematical Formula Ever" $$e^{i \pi}+1=0$$ is almost The Most Beautiful Mathematical Formula Ever.
Is it so?
Thanks in advance.
Rather than go through a two step definition of $e^z$, you could define it as $$\sum_{n=0}^\infty \frac{z^n} {n!} $$ Thus the definition $e^{i\theta} =\cos\theta+i\sin\theta$ is in no way arbitrary, it is in fact the only way you can extend the real exponential $e^x$ to the complex plane and have it be differentiable.
If you plug $1$ into the uniform formula you get Euler's constant, and if you plug in $i\pi$ you get $-1$. Seems pretty natural to me.